DOCSLIB.ORG

COMPLEX HADAMARD MATRICES and APPLICATIONS Introduction 1

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
COMPLEX HADAMARD MATRICES and APPLICATIONS Introduction 1 COMPLEX HADAMARD MATRICES AND APPLICATIONS TEO BANICA Abstract. A complex Hadamard matrix is a square matrix H 2 MN (C) whose entries are on the unit circle, jHijj = 1, and whose rows and pairwise orthogonal. The main ij 2πi=N example is the Fourier matrix, FN = (w ) with w = e . We discuss here the basic theory of such matrices, with emphasis on geometric and analytic aspects. CONTENTS Introduction 1 1. Hadamard matrices 5 2. Complex matrices 21 3. Roots of unity 37 4. Geometry, defect 53 5. Special matrices 69 6. Circulant matrices 85 7. Bistochastic matrices 101 8. Glow computations 117 9. Norm maximizers 133 10. Quantum groups 149 11. Subfactor theory 165 12. Fourier models 181 References 197 Introduction A complex Hadamard matrix is a square matrix H 2 MN (C) whose entries belong to the unit circle in the complex plane, Hij 2 T, and whose rows are pairwise orthogonal N P with respect to the usual scalar product of C , given by < x; y >= i xiy¯i. 2010 Mathematics Subject Classification. 15B10. Key words and phrases. Hadamard matrix, Fourier matrix. 1 2 TEO BANICA p The orthogonality condition tells us that the rescaled matrix U = H= N must be unitary. Thus, these matrices form a real algebraic manifold, given by: p XN = MN (T) \ NUN ij 2πi=N The basic example is the Fourier matrix, FN = (w ) with w = e . In standard matrix form, and with indices i; j = 0; 1;:::;N − 1, this matrix is as follows: 01 1 1 ::: 1 1 B1 w w2 : : : wN−1 C B C B1 w2 w4 : : : w2(N−1)C FN = B C B. C @. A 1 wN−1 w(2N−1) : : : w(N−1)2 More generally, we have as example the Fourier coupling of any finite abelian group G, regarded via the isomorphism G ' Gb as a square matrix, FG 2 MG(C): F =< i; j > G i2G;j2Gb Observe that for the cyclic group G = ZN we obtain in this way the above standard Fourier matrix FN . In general, we obtain a tensor product of Fourier matrices FN . There are many other examples of such matrices, for the most coming from various combinatorial constructions, basically involving design theory, and roots of unity. In addition, there are several deformation procedures for such matrices, leading to some more complicated constructions as well, or real algebraic geometry flavor. In general, the complex Hadamard matrices can be thought of as being \generalized Fourier matrices", of somewhat exotic type. Due to their generalized Fourier nature, these matrices appear in a wide array of questions in mathematics and physics: 1. Operator algebras. One important concept in the theory of von Neumann algebras is that of maximal abelian subalgebra (MASA). In the finite case, where the algebra has a trace, one can talk about pairs of orthogonal MASA. In the simplest case, of the matrix ∗ algebra MN (C), the orthogonal MASA are, up to conjugation, A = ∆;B = H∆H , where ∆ ⊂ MN (C) are the diagonal matrices, and H 2 MN (C) is Hadamard. 2. Subfactor theory. Along the same lines, but at a more advanced level, associated ∗ to any Hadamard matrix H 2 MN (C) is the square diagram C ⊂ ∆;H∆H ⊂ MN (C) formed by the associated MASA, which is a commuting square in the sense of subfactor theory. The Jones basic construction produces, out of this diagram, an index N subfactor of the Murray-von Neumann factor R, whose computation a key problem. HADAMARD MATRICES 3 3. Quantum groups. Associated to any complex Hadamard matrix H 2 MN (C) is a + certain quantum permutation group G ⊂ SN , obtained by factorizing the flat representa- + tion π : C(SN ) ! MN (C) associated to H. As a basic example here, the Fourier matrix FG produces in this way the group G itself. In general, the above-mentioned subfactor can be recovered from G, whose computation is a key problem. 4. Lattice models. According to the work of Jones, the combinatorics of the subfactor associated to an Hadamard matrix H 2 MN (C), which by the above can be recovered + from the representation theory of the associated quantum permutation group G ⊂ SN , can be thought of as being the combinatorics of a \spin model", in the context of link invariants, or of statistical mechanics, in an abstract, mathematical sense. From a more applied point of view, the Hadamard matrices can be used in order to construct mutually unbiased bases (MUB) and other useful objects, which can help in connection with quantum information theory, and other quantum physics questions. All this is quite recent, basically going back to the 00s. Regarding the known facts about the Hadamard matrices, most of them are in fact of purely mathematical nature. There are indeed many techniques that can be applied, leading to various results: 1. Algebra. In the real case, H 2 MN (±1), the study of such matrices goes back to the beginning of the 20th century, and is quite advanced. The main problems, however, namely the Hadamard conjecture (HC) and the circulant Hadamard conjecture (CHC) are not solved yet, with no efficient idea of approach in sight. Part of the real matrix techniques apply quite well to the root of unity case, H 2 MN (Zs), with s < 1. 2. Geometry. As already explained above,p the N × N complex Hadamard matrices form a real algebraic manifold, XN = MN (T) \ NUN . This manifold is highly singular, but several interesting geometric results about it have been obtained, notably about the general structure of the singularity at a given point H 2 XN , about the neighborhood of the Fourier matrices FG, and about the various isolated points as well. 3. Analysis. One interesting point of view on the Hadamard matrices, realp or complex, comes from the fact that these are precisely the rescaled versions, H = NU, of the P matrices which maximize the 1-norm jjUjj1 = ij jUijj on ON ;UN respectively. When looking more generally at the local maximizers of the 1-norm, one is led into a notion of \almost Hadamard matrices", having interesting algebraic and analytic aspects. 4. Probability. Another speculative approach, this time probabilistic, is by playing a Gale-Berlekamp type game with the matrix, in the hope that the invariants which are obtained in this way are related to the various geometric and quantum algebraic invariants, 4 TEO BANICA which are hard to compute. All this is related to the subtle fact that any unitary matrix, and so any complex Hadamard matrix as well, can be put in bistochastic form. Our aim here is to survey this material, theory and applications. Organizing all this is not easy, and we have chosen an algebra/geometry/analysis/physics lineup for our presentation, vaguely coming from the amount of background which is needed. The present text is organized in 4 parts, as follows: (1) Sections 1-3 contain basic definitions and various algebraic results. (2) Sections 4-6 deal with differential and algebraic geometric aspects. (3) Sections 7-9 are concerned with various analytic considerations. (4) Sections 10-12 deal with various mathematical physics aspects. There are of course many aspects of the theory which are missing from our presentation, but we will provide of course some information here, comments and references. Acknowledgements. I would like to thank Vaughan Jones for suggesting me, back to a discussion that we had in 1997, when we first met, to look at vertex models, and related topics. Stepping into bare Hadamard matrices is quite an experience, and very inspiring was the work of Uffe Haagerup on the subject, and his papers [32], [51], [52]. The present text is heavily based on a number of research papers on the subject that I wrote or co-signed, mostly during 2005{2015, and I would like to thank my cowork- ers Julien Bichon, Beno^ıtCollins, Ion Nechita, Remus Nicoar˘a,Duygu Ozteke,¨ Lorenzo Pittau, Jean-Marc Schlenker, Adam Skalski and Karol Zyczkowski._ Finally, many thanks go to my cats, for advice with hunting techniques, martial arts, and more. When doing linear algebra, all this knowledge is very useful. HADAMARD MATRICES 5 1. Hadamard matrices We are interested here in the complex Hadamard matrices, but we will start with some beautiful pure mathematics, regarding the real case. The definition that we need, going back to 19th century work of Sylvester [85], is as follows: Definition 1.1. An Hadamard matrix is a square binary matrix, H 2 MN (±1) whose rows are pairwise orthogonal, with respect to the scalar product on RN . As a first observation, we do not really need real numbers in order to talk about the Hadamard matrices, because the orthogonality condition tells us that, when comparing two rows, the number of matchings should equal the number of mismatchings. Thus, we can replace if we want the 1; −1 entries of our matrix by any two symbols, of our choice. Here is an example of an Hadamard matrix, with this convention: ~ ~ ~ ~ ~ | ~ | ~ ~ | | ~ | | ~ However, it is probably better to run away from this, and use real numbers instead, as in Definition 1.1, with the idea in mind of connecting the Hadamard matrices to the foundations of modern mathematics, namely Calculus 1 and Calculus 2. So, getting back now to the real numbers, here is a first result: Proposition 1.2. The set of the N × N Hadamard matrices is p YN = MN (±1) \ NON where ON is the orthogonal group, the intersection being taken inside MN (R). p Proof. Let H 2 MN (±1). Since the rows of the rescaled matrix U = H= N have norm 1, with respect to the usual scalar product on RN , we conclude that H is Hadamard precisely when U belongs to the orthogonal group ON , and so when H 2 YN , as claimed.
Load more

Recommended publications
  • A Finiteness Result for Circulant Core Complex Hadamard Matrices
    A FINITENESS RESULT FOR CIRCULANT CORE COMPLEX HADAMARD MATRICES REMUS NICOARA AND CHASE WORLEY UNIVERSITY OF TENNESSEE, KNOXVILLE Abstract. We show that for every prime number p there exist finitely many circulant core complex Hadamard matrices of size p ` 1. The proof uses a 'derivative at infinity' argument to reduce the problem to Tao's uncertainty principle for cyclic groups of prime order (see [Tao]). 1. Introduction A complex Hadamard matrix is a matrix H P MnpCq having all entries of absolute value 1 and all rows ?1 mutually orthogonal. Equivalently, n H is a unitary matrix with all entries of the same absolute value. For ij 2πi{n example, the Fourier matrix Fn “ p! q1¤i;j¤n, ! “ e , is a Hadamard matrix. In the recent years, complex Hadamard matrices have found applications in various topics of mathematics and physics, such as quantum information theory, error correcting codes, cyclic n-roots, spectral sets and Fuglede's conjecture. A general classification of real or complex Hadamard matrices is not available. A catalogue of most known complex Hadamard matrices can be found in [TaZy]. The complete classification is known for n ¤ 5 ([Ha1]) and for self-adjoint matrices of order 6 ([BeN]). Hadamard matrices arise in operator algebras as construction data for hyperfinite subfactors. A unitary ?1 matrix U is of the form n H, H Hadamard matrix, if and only if the algebra of n ˆ n diagonal matrices Dn ˚ is orthogonal onto UDnU , with respect to the inner product given by the trace on MnpCq. Equivalently, the square of inclusions: Dn Ă MnpCq CpHq “ ¨ YY ; τ˛ ˚ ˚ ‹ ˚ C Ă UDnU ‹ ˚ ‹ ˝ ‚ is a commuting square, in the sense of [Po1],[Po2], [JS].
    [Show full text]
  • Binary Polynomial Transforms for Nonlinear Signal Processing
    BINARY POLYNOMIAL TRANSFORMS FOR NONLINEAR SIGNAL PROCESSING Jaakko Astola, Karen Egiazarian, Rugen oktern SOS Agaian Signal Processing Laboratory Electrical Engineering Department Tampere University of Technology University of Texas at San Antonio Tampere, FINLAND San Antonio, Texas, USA ABSTRACT ing computers or systems suitable for simultaneous arith- metical or logical operations. In this paper we introduce parametric binary Rademacher functions of two types. Based on them using different kinds 2. BINARY POLYNOMIAL FUNCTIONS AND of arithmetical and logical operations we generate a set of MATRICES binary polynomial transforms (BPT). Some applications of BPT in nonlinear filtering and in compression of binary 2.1. Rademacher Functions and Matrices images are presented. Let a and b be arbitrary integers. We form the classes r,(t,a, b) and s,(t,a, b) of real periodic functions in the 1. INTRODUCTION following way. Let Spectral analysis is a powerful tool in communication, sig- ro(t.a,b) E so(t,a, b) E 1, (1) nal/ image processing and applied mathematics and there are many reasons behind the usefulness of spectral analysis. First, in many applications it is convenient to transform a a, fortEU~~ol[$,fk++i), problem into another, easier problem; the spectral domain rn+l (t, a, b) = is essentially a steady-state viewpoint; this not only gives b, for t E U~~ol[ $t + 5&l *).2 insight for analysis but also for design. Second, and more (2) important: the spectral approach allows us to treat entire and classes of signals which have similar properties in the spec- tral domain; it has excellent properties such as fast algo- a: for t E LO,&I, n = 0, 1,2, .
    [Show full text]
  • Hadamard Equiangular Tight Frames
    Air Force Institute of Technology AFIT Scholar Faculty Publications 8-8-2019 Hadamard Equiangular Tight Frames Matthew C. Fickus Air Force Institute of Technology John Jasper University of Cincinnati Dustin G. Mixon Air Force Institute of Technology Jesse D. Peterson Air Force Institute of Technology Follow this and additional works at: https://scholar.afit.edu/facpub Part of the Discrete Mathematics and Combinatorics Commons Recommended Citation Fickus, M., Jasper, J., Mixon, D. G., & Peterson, J. D. (2021). Hadamard Equiangular Tight Frames. Applied and Computational Harmonic Analysis, 50, 281–302. https://doi.org/10.1016/j.acha.2019.08.003 This Article is brought to you for free and open access by AFIT Scholar. It has been accepted for inclusion in Faculty Publications by an authorized administrator of AFIT Scholar. For more information, please contact [email protected]. Hadamard Equiangular Tight Frames Matthew Fickusa, John Jasperb, Dustin G. Mixona, Jesse D. Petersona aDepartment of Mathematics and Statistics, Air Force Institute of Technology, Wright-Patterson AFB, OH 45433 bDepartment of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45221 Abstract An equiangular tight frame (ETF) is a type of optimal packing of lines in Euclidean space. They are often represented as the columns of a short, fat matrix. In certain applications we want this matrix to be flat, that is, have the property that all of its entries have modulus one. In particular, real flat ETFs are equivalent to self-complementary binary codes that achieve the Grey-Rankin bound. Some flat ETFs are (complex) Hadamard ETFs, meaning they arise by extracting rows from a (complex) Hadamard matrix.
    [Show full text]
  • The Early View of “Global Journal of Computer Science And
    The Early View of “Global Journal of Computer Science and Technology” In case of any minor updation/modification/correction, kindly inform within 3 working days after you have received this. Kindly note, the Research papers may be removed, added, or altered according to the final status. Global Journal of Computer Science and Technology View Early Global Journal of Computer Science and Technology Volume 10 Issue 12 (Ver. 1.0) View Early Global Academy of Research and Development © Global Journal of Computer Global Journals Inc. Science and Technology. (A Delaware USA Incorporation with “Good Standing”; Reg. Number: 0423089) Sponsors: Global Association of Research 2010. Open Scientific Standards All rights reserved. Publisher’s Headquarters office This is a special issue published in version 1.0 of “Global Journal of Medical Research.” By Global Journals Inc. Global Journals Inc., Headquarters Corporate Office, All articles are open access articles distributed Cambridge Office Center, II Canal Park, Floor No. under “Global Journal of Medical Research” 5th, Cambridge (Massachusetts), Pin: MA 02141 Reading License, which permits restricted use. Entire contents are copyright by of “Global United States Journal of Medical Research” unless USA Toll Free: +001-888-839-7392 otherwise noted on specific articles. USA Toll Free Fax: +001-888-839-7392 No part of this publication may be reproduced Offset Typesetting or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information Global Journals Inc., City Center Office, 25200 storage and retrieval system, without written permission. Carlos Bee Blvd. #495, Hayward Pin: CA 94542 The opinions and statements made in this United States book are those of the authors concerned.
    [Show full text]
  • Asymptotic Existence of Hadamard Matrices
    ASYMPTOTIC EXISTENCE OF HADAMARD MATRICES by Ivan Livinskyi A Thesis submitted to the Faculty of Graduate Studies of The University of Manitoba in partial fulfilment of the requirements of the degree of MASTER OF SCIENCE Department of Mathematics University of Manitoba Winnipeg Copyright ⃝c 2012 by Ivan Livinskyi Abstract We make use of a structure known as signed groups, and known sequences with zero autocorrelation to derive new results on the asymptotic existence of Hadamard matrices. For any positive odd integer p it is obtained that a Hadamard matrix of order 2tp exists for all ( ) 1 p − 1 t ≥ log + 13: 5 2 2 i Contents 1 Introduction. Hadamard matrices 2 2 Signed groups and remreps 10 3 Construction of signed group Hadamard matrices from sequences 18 4 Some classes of sequences with zero autocorrelation 28 4.1 Golay sequences .............................. 30 4.2 Base sequences .............................. 31 4.3 Complex Golay sequences ........................ 33 4.4 Normal sequences ............................. 35 4.5 Turyn sequences .............................. 36 4.6 Other sequences .............................. 37 5 Asymptotic Existence results 40 5.1 Asymptotic formulas based on Golay sequences . 40 5.2 Hadamard matrices from complex Golay sequences . 43 5.3 Hadamard matrices from all complementary sequences considered . 44 6 Thoughts about further development 48 Bibliography 51 1 Chapter 1 Introduction. Hadamard matrices A Hadamard matrix H is a square (±1)-matrix such that any two of its rows are orthogonal. In other words, a square (±1)-matrix H is Hadamard if and only if HH> = nI; where n is the order of H and I is the identity matrix of order n.
    [Show full text]
  • LU Decomposition - Wikipedia, the Free Encyclopedia
    LU decomposition - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/LU_decomposition From Wikipedia, the free encyclopedia In linear algebra, LU decomposition (also called LU factorization) is a matrix decomposition which writes a matrix as the product of a lower triangular matrix and an upper triangular matrix. The product sometimes includes a permutation matrix as well. This decomposition is used in numerical analysis to solve systems of linear equations or calculate the determinant of a matrix. LU decomposition can be viewed as a matrix form of Gaussian elimination. LU decomposition was introduced by mathematician Alan Turing [1] 1 Definitions 2 Existence and uniqueness 3 Positive definite matrices 4 Explicit formulation 5 Algorithms 5.1 Doolittle algorithm 5.2 Crout and LUP algorithms 5.3 Theoretical complexity 6 Small example 7 Sparse matrix decomposition 8 Applications 8.1 Solving linear equations 8.2 Inverse matrix 8.3 Determinant 9 See also 10 References 11 External links Let A be a square matrix. An LU decomposition is a decomposition of the form where L and U are lower and upper triangular matrices (of the same size), respectively. This means that L has only zeros above the diagonal and U has only zeros below the diagonal. For a matrix, this becomes: An LDU decomposition is a decomposition of the form where D is a diagonal matrix and L and U are unit triangular matrices, meaning that all the entries on the diagonals of L and U LDU decomposition of a Walsh matrix 1 of 7 1/11/2012 5:26 PM LU decomposition - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/LU_decomposition are one.
    [Show full text]
  • One Method for Construction of Inverse Orthogonal Matrices
    ONE METHOD FOR CONSTRUCTION OF INVERSE ORTHOGONAL MATRICES, P. DIÞÃ National Institute of Physics and Nuclear Engineering, P.O. Box MG6, Bucharest, Romania E-mail: [email protected] Received September 16, 2008 An analytical procedure to obtain parametrisation of complex inverse orthogonal matrices starting from complex inverse orthogonal conference matrices is provided. These matrices, which depend on nonzero complex parameters, generalize the complex Hadamard matrices and have applications in spin models and digital signal processing. When the free complex parameters take values on the unit circle they transform into complex Hadamard matrices. 1. INTRODUCTION In a seminal paper [1] Sylvester defined a general class of orthogonal matrices named inverse orthogonal matrices and provided the first examples of what are nowadays called (complex) Hadamard matrices. A particular class of inverse orthogonal matrices Aa()ij are those matrices whose inverse is given 1 t by Aa(1ij ) (1 a ji ) , where t means transpose, and their entries aij satisfy the relation 1 AA nIn (1) where In is the n-dimensional unit matrix. When the entries aij take values on the unit circle A–1 coincides with the Hermitian conjugate A* of A, and in this case (1) is the definition of complex Hadamard matrices. Complex Hadamard matrices have applications in quantum information theory, several branches of combinatorics, digital signal processing, etc. Complex orthogonal matrices unexpectedly appeared in the description of topological invariants of knots and links, see e.g. Ref. [2]. They were called two- weight spin models being related to symmetric statistical Potts models. These matrices have been generalized to two-weight spin models, also called , Paper presented at the National Conference of Physics, September 10–13, 2008, Bucharest-Mãgurele, Romania.
    [Show full text]
  • Introduction to Linear Algebra
    Introduction to linear algebra Teo Banica Real matrices, The determinant, Complex matrices, Calculus and applications, Infinite matrices, Special matrices 08/20 Foreword These are slides written in the Fall 2020, on linear algebra. Presentations available at my Youtube channel. 1. Real matrices and their properties ... 3 2. The determinant of real matrices ... 19 3. Complex matrices and diagonalization ... 35 4. Linear algebra and calculus questions ... 51 5. Infinite matrices and spectral theory ... 67 6. Special matrices and matrix tricks ... 83 Real matrices and their properties Teo Banica "Introduction to linear algebra", 1/6 08/20 Rotations 1/3 Problem: what’s the formula of the rotation of angle t? Rotations 2/3 2 x The points in the plane R can be represented as vectors y . The 2 × 2 matrices “act” on such vectors, as follows: a b x ax + by = c d y cx + dy Many simple transformations (symmetries, projections..) can be written in this form. What about the rotation of angle t? Rotations 3/3 A quick picture shows that we must have: ∗ ∗ 1 cos t = ∗ ∗ 0 sin t Also, by paying attention to positives and negatives: ∗ ∗ 0 − sin t = ∗ ∗ 1 cos t Thus, the matrix of our rotation can only be: cos t − sin t R = t sin t cos t By "linear algebra”, this is the correct answer. Linear maps 1/4 2 2 Theorem. The maps f : R ! R which are linear, in the sense that they map lines through 0 to lines through 0, are: x ax + by f = y cx + dy Remark. If we make the multiplication convention a b x ax + by = c d y cx + dy the theorem says f (v) = Av, with A being a 2 × 2 matrix.
    [Show full text]
  • Perron Spectratopes and the Real Nonnegative Inverse Eigenvalue Problem
    Perron Spectratopes and the Real Nonnegative Inverse Eigenvalue Problem a b, Charles R. Johnson , Pietro Paparella ∗ aDepartment of Mathematics, College of William & Mary, Williamsburg, VA 23187-8795, USA bDivision of Engineering and Mathematics, University of Washington Bothell, Bothell, WA 98011-8246, USA Abstract Call an n-by-n invertible matrix S a Perron similarity if there is a real non- 1 scalar diagonal matrix D such that SDS− is entrywise nonnegative. We give two characterizations of Perron similarities and study the polyhedra (S) := n 1 C x R : SDxS− 0;Dx := diag (x) and (S) := x (S): x1 = 1 , whichf 2 we call the Perron≥ spectracone andgPerronP spectratopef 2, respectively. C Theg set of all normalized real spectra of diagonalizable nonnegative matrices may be covered by Perron spectratopes, so that enumerating them is of interest. The Perron spectracone and spectratope of Hadamard matrices are of par- ticular interest and tend to have large volume. For the canonical Hadamard matrix (as well as other matrices), the Perron spectratope coincides with the convex hull of its rows. In addition, we provide a constructive version of a result due to Fiedler ([9, Theorem 2.4]) for Hadamard orders, and a constructive version of [2, Theorem 5.1] for Sule˘ımanova spectra. Keywords: Perron spectracone, Perron spectratope, real nonnegative inverse eigenvalue problem, Hadamard matrix, association scheme, relative gain array 2010 MSC: 15A18, 15B48, 15A29, 05B20, 05E30 1. Introduction The real nonnegative inverse eigenvalue problem (RNIEP) is to determine arXiv:1508.07400v2 [math.RA] 19 Nov 2015 which sets of n real numbers occur as the spectrum of an n-by-n nonnegative matrix.
    [Show full text]
  • The 2-Transitive Complex Hadamard Matrices
    undergoing revision for Designs, Codes and Cryptography 1 January, 2001 The 2-Transitive Complex Hadamard Matrices G. Eric Moorhouse Dept. of Mathematics, University of Wyoming, Laramie WY, U.S.A. Abstract. We determine all possibilities for a complex Hadamard ma- trix H admitting an automorphism group which permutes 2-transitively the rows of H. Our proof of this result relies on the classification theo- rem for finite 2-transitive permutation groups, and thereby also on the classification of finite simple groups. Keywords. complex Hadamard matrix, 2-transitive, distance regular graph, cover of complete bipartite graph 1. Introduction Let H be a complex Hadamard matrix of order v, i.e. a v × v matrix whose entries are com- plex roots of unity, satisfying HH∗ = vI,where∗ denotes conjugate-transpose. Butson’s original definition in [3] considered as entries only p-th roots of unity for some prime p, while others [42], [32], [8] have considered ±1, ±i as entries. These are generalisations of the (ordinary) Hadamard matrices (having entries ±1); other generalisations with group entries are described in Section 2. 1.1 Example. The complex Hadamard matrix ⎡ ⎤ 111111 ⎢ 2 2 ⎥ ⎢ 11 ωω ω ω ⎥ ⎢ 2 2 ⎥ ⎢ 1 ω 1 ωω ω ⎥ 2πi/3 H6 = ⎢ 2 2 ⎥ ,ω= e ⎣ 1 ω ω 1 ωω⎦ 1 ω2 ω2 ω 1 ω 1 ωω2 ω2 ω 1 of order 6 corresponds (as in Section 2) to the distance transitive triple cover of the complete bipartite graph K6,6 which appears in [2, Thm. 13.2.2]. ∗ An automorphism of H is a pair (M1 ,M2) of monomial matrices such that M1HM2 = H.
    [Show full text]
  • On Quaternary Complex Hadamard Matrices of Small Orders
    Advances in Mathematics of Communications doi:10.3934/amc.2011.5.309 Volume 5, No. 2, 2011, 309{315 ON QUATERNARY COMPLEX HADAMARD MATRICES OF SMALL ORDERS Ferenc Szoll¨ osi} Department of Mathematics and its Applications Central European University H-1051, N´adoru. 9, Budapest, Hungary (Communicated by Gilles Z´emor) Abstract. One of the main goals of design theory is to classify, character- ize and count various combinatorial objects with some prescribed properties. In most cases, however, one quickly encounters a combinatorial explosion and even if the complete enumeration of the objects is possible, there is no ap- parent way how to study them in details, store them efficiently, or generate a particular one rapidly. In this paper we propose a novel method to deal with these difficulties, and illustrate it by presenting the classification of quaternary complex Hadamard matrices up to order 8. The obtained matrices are mem- bers of only a handful of parametric families, and each inequivalent matrix, up to transposition, can be identified through its fingerprint. 1. Introduction A complex Hadamard matrix H of order n is an n × n matrix with unimodular entries satisfying HH∗ = nI where ∗ is the conjugate transpose and I is the iden- tity matrix of order n. In other words, any two distinct rows (or columns) of H are complex orthogonal. Complex Hadamard matrices have important applications in quantum optics, high-energy physics [1], operator theory [16] and in harmonic analysis [12], [20]. They also play a crucial r^olein quantum information theory, for construction of teleportation and dense coding schemes [21], and they are strongly related to mutually unbiased bases (MUBs) [8].
    [Show full text]
  • Almost Hadamard Matrices with Complex Entries
    Adv. Oper. Theory 3 (2018), no. 1, 137–177 http://doi.org/10.22034/aot.1702-1114 ISSN: 2538-225X (electronic) http://aot-math.org ALMOST HADAMARD MATRICES WITH COMPLEX ENTRIES TEODOR BANICA,1∗ and ION NECHITA2 Dedicated to the memory of Uffe Haagerup Communicated by N. Spronk Abstract. We discuss an extension of the almost Hadamard matrix formal- ism, to the case of complex matrices. Quite surprisingly, the situation here is very different from the one in the real case, and our conjectural conclusion is that there should be no such matrices, besides the usual Hadamard ones. We verify this conjecture in a number of situations, and notably for most of the known examples of real almost Hadamard matrices, and for some of their complex extensions. We discuss as well some potential applications of our conjecture, to the general study of complex Hadamard matrices. Introduction An Hadamard matrix is a square matrix H ∈ MN (±1), whose rows are pairwise orthogonal. Here is a basic example, which appears as a version of the 4×4 Walsh matrix, and is also well-known for its use in the Grover algorithm: −1 1 1 1 1 −1 1 1 K4 = 1 1 −1 1 1 1 1 −1 Assuming that the matrix has N ≥ 3 rows, the orthogonality conditions be- tween the rows give N ∈ 4N. A similar analysis with four or more rows, or any Copyright 2016 by the Tusi Mathematical Research Group. Date: Received: Feb. 9, 2017; Accepted: May 12, 2017. ∗Corresponding author. 2010 Mathematics Subject Classification. Primary 15B10; Secondary 05B20, 14P05.
    [Show full text]